Theorem: Cycle velocities and finite-time offset

Let $q=|V(Q)|$ and $D_{\max }={\max }_e\Vert {\Delta }_e{\Vert }_2$. Every dependency path completed within time $u$ satisfies

The set of limiting path velocities as elapsed time tends to infinity is exactly $K_{\mathsf{Sch}}$. This is a statement about potential scheduled reachability, not transmission by every path.

Proof. Remove directed cycles from a finite walk until the remainder has no repeated vertex. Decompose the removed cycles into simple cycles. The remainder has at most $q-1$ edges, so its displacement has norm at most $(q-1)D_{\max }$.

The cycle contribution is

Its total duration is no greater than $u$. Since $0\in K_{\mathsf{Sch}}$, the contribution belongs to $u{K}_{\mathsf{Sch}}$, proving (eq:finiteoffset). Dividing by elapsed time proves the upper inclusion for limiting velocities.

Conversely, approximate a convex combination of cycle velocities and $0$ by spending the corresponding time proportions on repeated cycles and waiting. Strong connectivity supplies fixed connector walks between their base vertices. Choose repetition numbers tending to infinity; connector durations and displacements become negligible. Integer repetition numbers approximate the desired time proportions, giving every velocity in $K_{\mathsf{Sch}}$.